Pure electron or pure positron plasmas held in magnetic fields B radiate energy because of the cyclotron motion of the plasma particles; nominally, the plasmas should cool to the often cryogenic temperatures of the trap in which they are confined. However, the cyclotron cooling rate for leptons is (1/4 s)(B/1 T)2, and significant cooling is not normally observed unless B≳1 T. Cooling to the trap temperatures of ∼10 K is particularly difficult to attain. Here, we show that dramatically higher cooling rates (×100) and lower temperatures (÷1000) can be obtained if the plasmas are held in electromagnetic cavities rather than in effectively free space conditions. We find that plasmas with up to 107 particles can be cooled in fields close to 0.15 T, much lower than 1 T commonly thought to be necessary to obtain plasma cooling. Appropriate cavities can be constructed with only minor modifications to the standard Penning-Malmberg trap structures.